OCR FP1 2012 June — Question 5 5 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve summation with exponentials
DifficultyStandard +0.3 This is a straightforward proof by induction with a summation involving exponentials. While it's a Further Maths topic (making it slightly above average), the structure is completely standard: verify base case n=1, assume for n=k, prove for n=k+1 by adding the next term and factoring out 3^k. The algebra is routine and requires no novel insight—just careful manipulation of geometric series terms.
Spec4.01a Mathematical induction: construct proofs

5 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } 4 \times 3 ^ { r } = 6 \left( 3 ^ { n } - 1 \right)\).

Question 5:
AnswerMarks
B1Verify result true when \(n = 1\)
M1*Add next term in series
DepM1Attempt to obtain \(3^{k+1}\) correctly
A1Show sufficient working to justify correct expression
B1Clear statements of Induction processes, but first 4 marks must all be earned
[5]
## Question 5:

| B1 | Verify result true when $n = 1$
| M1* | Add next term in series
| DepM1 | Attempt to obtain $3^{k+1}$ correctly
| A1 | Show sufficient working to justify correct expression
| B1 | Clear statements of Induction processes, but first 4 marks must all be earned
**[5]**

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5 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } 4 \times 3 ^ { r } = 6 \left( 3 ^ { n } - 1 \right)$.

\hfill \mbox{\textit{OCR FP1 2012 Q5 [5]}}